A finite volume scheme for nonlinear degenerate parabolic equations
For researchers working on numerical methods for degenerate parabolic PDEs, this scheme offers improved long-time behavior and accuracy, though it is an incremental improvement over existing finite volume methods.
The authors propose a second-order finite volume scheme for nonlinear degenerate parabolic equations that preserves steady-states and provides accurate long-time behavior, with numerical results confirming second-order accuracy in both degenerate and non-degenerate regimes.
We propose a second order finite volume scheme for nonlinear degenerate parabolic equations. For some of these models (porous media equation, drift-diffusion system for semiconductors, ...) it has been proved that the transient solution converges to a steady-state when time goes to infinity. The present scheme preserves steady-states and provides a satisfying long-time behavior. Moreover, it remains valid and second-order accurate in space even in the degenerate case. After describing the numerical scheme, we present several numerical results which confirm the high-order accuracy in various regime degenerate and non degenerate cases and underline the efficiency to preserve the large-time asymptotic.